Uniform Continuity (Example 1): The Basics

[@hypershadic5093]

Thank you so much for reviewing this subject of analysis, I was really struggling with these definitions!

[@dylank6191]

The best thing about uniform continuity is that it's for free for every continuous function defined on a compact space. So, as a little fun fact, f(x) = x^2 is uniformly continuous on every intervall [a,b] (and every other bounded and closed set for that matter)

[@呂永志]

So is every polynomial, I think.

[@dylank6191]

@@呂永志 Yep, since every polynomial is continuous.

[@arturcostasteiner9735]

I think it's interesting to notice that every Lipschitz function is uniformly continuous but the converse is not true.

[@tofu8676]

yes i usually associate LC with UC so it would be very interesting to see such a function!

[@wtt274]

Sir has given a perfect explanation on uniform continuity in his great video .Thank you sir !

[@ABC-jq7ve]

I love your videos!!

[@tomasasnes4081]

I actually understood this!!! Awsome video

[@arturcostasteiner9735]

On the complex domain, there's an amazing conclusion, which doesn't seem to be very known: If f is entire and uniformly continuous on C, then f is an affine mapping: for every z, f(z) = az + b, a and b complex constants.

[@bijnas0074]

Love from Nepal sir .🇳🇵

[@andrewmichel2525]

I will take real analysis in the near future and am a bit worried about it but these videos make me feel like the class wont completely blindside me

[@ranahalide9422]

Thanks sir from india ❤️

[@asamenechbayissa553]

I am getting interested in analysis from those videos

[@蔺美云]

Awesome!

[@tomoki-v6o]

for my engineering problems i use the perpondicular to the bisector angle between left and right tangents and i trait it as a derivative

[@Scrollwithvicky333]

Love from 🇮🇳 india❤

[@alfredocuomo4284]

Great....

[@maximelamoureux9836]

Your video are really helpfull. Can you make a video on several example to prove that a function isn't UC ?

[@drpeyam]

Check out the playlist 😉

[@DynamicMateTV]

Could you please make video explaining uniform continuity visually/diagrmatically?

[@drpeyam]

Already done, check the description

[@MarkusHedemann]

Excellent, thank you! However, I don't see why You are constrained to add 3 to 3. Why not justadd the absolute value of -1 to 3? That will give epsilon/4 which in the end gives the same result (which, of course, is a reason why you might use 3 as well as the absolute value of -1).

[@him21016]

You say you’ll cover in an upcoming video how functions on compact sets are uniformly continuous but I first learnt about both those terms from a video you made ages ago! We already have this from you. Just a thought, but the definition of finite covers and compact sets was hard to understand - why is (1,6] not compact and [1,6] compact?

[@drpeyam]

For (1,6] the family (1+1/n,7) is an open cover that has no finite subcover

[@him21016]

@@drpeyam Is the difference sir that the limit as n goes to infinity of 1+1/n can touch all the points on the open interval (1, but it could never be exactly one, so you can't call it a cover for [1 closed?

[@theunknownscientist3249]

Every continuous function on a compact interval is uniformly continous on that interval?! That seems soooo awesome, I can't believe it. Hmm, maybe I'll try a proof? Kinda hard tho.

[@drpeyam]

There’s a proof in my playlist :)

[@Apollorion]

.. say h(x, epsilon) is the value of delta for which |x-y|

[@drpeyam]

I thought it would be like that too, but it’s a bit more complicated than that, unfortunately

[@andikaajisaputra8093]

Still support u

[@pierreabbat6157]

How about f(x)=cube root of x on all R?

[@drpeyam]

I think it is, since the derivative is bounded. Check out the derivative video on the playlist

[@pierreabbat6157]

@@drpeyam No it isn't; the derivative goes to ∞ at 0. But it's still uniformly continuous.

[@drpeyam]

It’s uniformly continuous on [-2,2] since it’s compact, and also UC on [-oo,-1]u[1,oo) since the derivative is bounded there, and then you just patch things up

[@yoav613]

So x^2 is unif cont in any finite interval unless delta=e/inf counts😆

[@drpeyam]

Yep :)

[@drpeyam]

But interestingly it is not uniformly continuous on R

[@yoav613]

@@drpeyam yes i agree. From your example it is easy to see why this is not uc in R because we can not choose delta= e/inf but it is intresting that we can always expand our interval more and more but still x^2 is not uc in R

[@ethancheung1676]

i always wonder why we dont use the scapework as the proof. because usually in these kind of epsilon-delta proof, everything is bidirectional. if it is purely about the format of the proof (prefer top to bottom), i think we can do the scapework from bottom to top, voila!

[@drpeyam]

It may look like it’s bidirectional, but sometimes it isn’t, as in Example 3

[@cricketshorts3402]

2nd

[@loucifabdessalam1522]

فوليس

[@colleen9493]

Use the Chen Lu! Just kidding I have no idea what this is

[@stevenmellemans7215]

I don’t find this very clear, if my daughter or son would give this as an answer, they would loose a lot of marks on that question. Not by me but but their teacher :-) They should use stricter notation. For instance I didn’t see any .